Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks

Bourlard-Jospin, Maryse; Nicaise, Serge; Venel, Juliette

Confluentes Mathematici, Tome 7 (2015) no. 1, pp. 13-33.

We show that the solution of the two-dimensional Dirichlet problem set in a plane domain is the limit of the solutions of similar problems set on a sequence of one-dimensional networks as their size goes to zero. Roughly speaking this means that a membrane can be seen as the limit of rackets made of strings. For practical applications, we also show that the solutions of the discrete approximated problems (again on the one-dimensional networks) also converge to the solution of the two-dimensional Dirichlet problem.

Classification : 35R02, 35B40, 65N30

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